Recent research in inverse problems seeks to develop a mathematically coherent foundation for combining data-driven models, and in particular those based on deep learning, with domain-specific knowledge contained in physical–analytical models. The focus is on solving ill-posed inverse problems that are at the core of many challenging applications in the natural sciences, medicine and life sciences, as well as in engineering and industrial applications. This survey paper aims to give an account of some of the main contributions in data-driven inverse problems.

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Solving inverse problems using data-driven models

Undersampled magnetic resonance image (MRI) reconstruction is typically an ill-posed linear inverse task. The time and resource intensive computations require tradeoffs between accuracy and speed . In addition, state-of-the-art compressed sensing (CS) analytics are not cognizant of the image diagnostic quality . To address these challenges, we propose a novel CS framework that uses generative adversarial networks (GAN) to model the (low-dimensional) manifold of high-quality MR images. Leveraging a mixture of least-squares (LS) GANs and pixel-wise $\ell _{1}/\ell _{2}$ cost, a deep residual network with skip connections is trained as the generator that learns to remove the aliasing artifacts by projecting onto the image manifold. The LSGAN learns the texture details, while the $\ell _{1}/\ell _{2}$ cost suppresses high-frequency noise. A discriminator network, which is a multilayer convolutional neural network (CNN), plays the role of a perceptual cost that is then jointly trained based on high-quality MR images to score the quality of retrieved images. In the operational phase, an initial aliased estimate (e.g., simply obtained by zero-filling) is propagated into the trained generator to output the desired reconstruction. This demands a very low computational overhead. Extensive evaluations are performed on a large contrast-enhanced MR dataset of pediatric patients. Images rated by expert radiologists corroborate that GANCS retrieves higher quality images with improved fine texture details compared with conventional Wavelet-based and dictionary-learning-based CS schemes as well as with deep-learning-based schemes using pixel-wise training. In addition, it offers reconstruction times of under a few milliseconds, which are two orders of magnitude faster than the current state-of-the-art CS-MRI schemes.

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Deep Generative Adversarial Neural Networks for Compressive Sensing MRI

Over past several years, machine learning, or more generally artificial intelligence, has generated overwhelming research interest and attracted unprecedented public attention. As tomographic imaging researchers, we share the excitement from our imaging perspective [item 1) in the Appendix], and organized this special issue dedicated to the theme of “Machine learning for image reconstruction.” This special issue is a sister issue of the special issue published in May 2016 of this journal with the theme “Deep learning in medical imaging” [item 2) in the Appendix]. While the previous special issue targeted medical image processing/analysis, this special issue focuses on data-driven tomographic reconstruction. These two special issues are highly complementary, since image reconstruction and image analysis are two of the main pillars for medical imaging. Together we cover the whole workflow of medical imaging: from tomographic raw data/features to reconstructed images and then extracted diagnostic features/readings.

Image Reconstruction is a New Frontier of Machine Learning

Recent advances in deep learning for tomographic reconstructions have shown great potential to create accurate and high quality images with a considerable speed up. In this paper, we present a deep neural network that is specifically designed to provide high resolution 3-D images from restricted photoacoustic measurements. The network is designed to represent an iterative scheme and incorporates gradient information of the data fit to compensate for limited view artifacts. Due to the high complexity of the photoacoustic forward operator, we separate training and computation of the gradient information. A suitable prior for the desired image structures is learned as part of the training. The resulting network is trained and tested on a set of segmented vessels from lung computed tomography scans and then applied to in-vivo photoacoustic measurement data.

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Model-Based Learning for Accelerated, Limited-View 3-D Photoacoustic Tomography

In the presence of metal implants, metal artifacts are introduced to x-ray computed tomography CT images. Although a large number of metal artifact reduction (MAR) methods have been proposed in the past decades, MAR is still one of the major problems in clinical x-ray CT. In this paper, we develop a convolutional neural network (CNN)-based open MAR framework, which fuses the information from the original and corrected images to suppress artifacts. The proposed approach consists of two phases. In the CNN training phase, we build a database consisting of metal-free, metal-inserted and pre-corrected CT images, and image patches are extracted and used for CNN training. In the MAR phase, the uncorrected and pre-corrected images are used as the input of the trained CNN to generate a CNN image with reduced artifacts. To further reduce the remaining artifacts, water equivalent tissues in a CNN image are set to a uniform value to yield a CNN prior, whose forward projections are used to replace the metal-affected projections, followed by the FBP reconstruction. The effectiveness of the proposed method is validated on both simulated and real data. Experimental results demonstrate the superior MAR capability of the proposed method to its competitors in terms of artifact suppression and preservation of anatomical structures in the vicinity of metal implants.

Convolutional Neural Network Based Metal Artifact Reduction in X-Ray Computed Tomography

Compressive sensing (CS) has proved effective for tomographic reconstruction from sparsely collected data or under-sampled measurements, which are practically important for few-view computed tomography (CT), tomosynthesis, interior tomography, and so on. To perform sparse-data CT, the iterative reconstruction commonly uses regularizers in the CS framework. Currently, how to choose the parameters adaptively for regularization is a major open problem. In this paper, inspired by the idea of machine learning especially deep learning, we unfold the state-of-the-art “fields of experts”-based iterative reconstruction scheme up to a number of iterations for data-driven training, construct a learned experts’ assessment-based reconstruction network (LEARN) for sparse-data CT, and demonstrate the feasibility and merits of our LEARN network. The experimental results with our proposed LEARN network produces a superior performance with the well-known Mayo Clinic low-dose challenge data set relative to the several state-of-the-art methods, in terms of artifact reduction, feature preservation, and computational speed. This is consistent to our insight that because all the regularization terms and parameters used in the iterative reconstruction are now learned from the training data, our LEARN network utilizes application-oriented knowledge more effectively and recovers underlying images more favorably than competing algorithms. Also, the number of layers in the LEARN network is only 50, reducing the computational complexity of typical iterative algorithms by orders of magnitude.

/pdf/learn-learned-experts-assessment-based-reconstruction-2ht18l6tq0.pdf

LEARN: Learned Experts’ Assessment-Based Reconstruction Network for Sparse-Data CT

Compressive sensing (CS) has proved effective for tomographic reconstruction from sparsely collected data or under-sampled measurements, which are practically important for few-view CT, tomosynthesis, interior tomography, and so on. To perform sparse-data CT, the iterative reconstruction commonly use regularizers in the CS framework. Currently, how to choose the parameters adaptively for regularization is a major open problem. In this paper, inspired by the idea of machine learning especially deep learning, we unfold a state-of-the-art "fields of experts" based iterative reconstruction scheme up to a number of iterations for data-driven training, construct a Learned Experts' Assessment-based Reconstruction Network ("LEARN") for sparse-data CT, and demonstrate the feasibility and merits of our LEARN network. The experimental results with our proposed LEARN network produces a competitive performance with the well-known Mayo Clinic Low-Dose Challenge Dataset relative to several state-of-the-art methods, in terms of artifact reduction, feature preservation, and computational speed. This is consistent to our insight that because all the regularization terms and parameters used in the iterative reconstruction are now learned from the training data, our LEARN network utilizes application-oriented knowledge more effectively and recovers underlying images more favorably than competing algorithms. Also, the number of layers in the LEARN network is only 12, reducing the computational complexity of typical iterative algorithms by orders of magnitude.

Yunjin Chen

Papers

LEARN: Learned Experts’ Assessment-Based Reconstruction Network for Sparse-Data CT

LEARN: Learned Experts' Assessment-based Reconstruction Network for Sparse-data CT